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Abstract

We examine the projection approximation in the context of propagation-based phase contrast imaging using hard x-rays. Specifically, we consider the case of a cylinder or a rounded edge, as a simple model for the edges of many biological samples. The Argand-plane signature of a propagation-based phase contrast fringe from the edge of a cylinder is studied, and the evolution of this signature with propagation. This, along with experimental images obtained using a synchrotron source, reveals how propagation within the scattering volume is not fully described in the projection approximation's ray-based approach. This means that phase contrast fringes are underestimated by the projection approximation at a short object-to-detector propagation distance, namely a distance comparable to the free-space propagation within the volume. This failure of the projection approximation may become non-negligible in the detailed study of small anatomical features deep within a large body. Nevertheless, the projection approximation matches the exact solution for a larger propagation distance typical of those used in biomedical phase contrast imaging.

Phase contrast images show the projection approximation correctly predicts the fringes seen at sufficiently long propagation distances. Images taken at the upstream hutch of BL20XU, SPring-8, of a 3 mm diameter perspex cylinder at 25keV for a propagation of a) 50 cm and b) 100 cm. The observed profile is shown in black and the simulated in blue.

Argand plot of 0.5 Angstrom waves incident on a 3 mm cylinder, smoothed by a Gaussian of pixel size 0.18 micron, showing the introduction of multiple intense, wider fringes with propagation, viewing 1 micron either side of the interface.

Experimental phase contrast images show the projection approximation does not predict the fringes seen at very short propagation distances, but the simulated fringes become more accurate as the propagation distance increases. Images taken of the same 3 mm diameter perspex cylinder as Fig. 3, at 25keV for a propagation of a) contact, b) 1.5 mm, c) 3 mm. At these short propagation distances, the pixel size and point spread function prevent multiple fringes being resolved at the edge interface. The shorter propagation distance means that the signal to noise ratio is reduced. The observed profile is shown in black and that simulated using the projection approximation (then smoothed with detector characteristics as in Fig. 3) in blue.